---
tag: QQ3K6CZASD
kind: concept
name: Hypersimplex
sources:
  - wikipedia: en:44749312@r1335660501
    locator: Hypersimplex
    note: adopted
    mention: wikipedia://en/44749312
    sections: [Summary]
same_as:
  wikidata: Q25303757
  wikipedia: Hypersimplex
inventory: [6Y12J9WXWN]
---

## Summary

In polyhedral combinatorics, the **hypersimplex** $\Delta_{d,k}$ is a convex polytope that generalizes the simplex. It is determined by two integers $d$ and $k$, and is defined as the [[convex hull|0FQSDC2EQ2]] of the $d$-dimensional vectors whose coefficients consist of $k$ ones and $d-k$ zeros. Equivalently, $\Delta_{d,k}$ can be obtained by slicing the $d$-dimensional unit hypercube $[0,1]^d$ with the hyperplane of equation $x_1+\cdots+x_d=k$ and, for this reason, it is a $(d-1)$-dimensional polytope when $0<k<d$.